1. Find all critical numbers of the function
.
a. critical numbers:
*b. critical numbers:
c. critical numbers:
d. critical numbers:
e. no critical numbers
2. Find any critical numbers of the function
.
a. 0
b.
c.
d.
*e.
3. Locate the absolute extrema of the function
interval
.
on the closed
a. absolute maximum:
absolute minimum:
b. absolute maximum:
absolute minimum:
*c. absolute maximum:
absolute minimum:
d. absolute maximum:
absolute minimum:
e. absolute maximum:
absolute minimum:
4. Determine whether Rolle's Theorem can be applied to the function
on the closed interval [–1,3]. If Rolle's Theorem can be
applied, find all values of c in the open interval (–1,3) such that
*a. Rolle's Theorem applies; c = 1
b. Rolle's Theorem applies; c = 2
c. Rolle's Theorem applies; c = 0
d. Rolle's Theorem applies; c = –1
e. Rolle's Theorem does not apply
5. Determine whether the Mean Value Theorem can be applied to the function
on the closed interval [3,9]. If the Mean Value Theorem can be
applied, find all numbers c in the open interval (3,9) such that
.
*a. MVT applies; c = 6
b. MVT applies; c = 7
c. MVT applies; c = 4
d. MVT applies; c = 5
e. MVT applies; c = 8
6. A plane begins its takeoff at 2:00 P.M. on a 2200-mile flight. After 12.5
hours, the plane arrives at its destination. Explain why there are at least
two times during the flight when the speed of the plane is 100 miles per hour.
a. By the Mean Value Theorem, there is a time when the speed of the
plane must equal the average speed of 303 mi/hr. The speed was 100 mi/hr
when the plane was accelerating to 303 mi/hr and decelerating from 303
mi/hr.
b. By the Mean Value Theorem, there is a time when the speed of the
plane must equal the average speed of 152 mi/hr. The speed was 100 mi/hr
when the plane was accelerating to 152 mi/hr and decelerating from 152
mi/hr.
*c. By the Mean Value Theorem, there is a time when the speed of the
plane must equal the average speed of 88 mi/hr. The speed was 100 mi/hr
when the plane was accelerating to 88 mi/hr and decelerating from 88
mi/hr.
d. By the Mean Value Theorem, there is a time when the speed of the
plane must equal the average speed of 117 mi/hr. The speed was 100 mi/hr
when the plane was accelerating to 117 mi/hr and decelerating from 117
mi/hr.
e. By the Mean Value Theorem, there is a time when the speed of the
plane must equal the average speed of 176 mi/hr. The speed was 100 mi/hr
when the plane was accelerating to 176 mi/hr and decelerating from 176
mi/hr.
7. Find a function f that has derivative
passing through the point (5,6).
and with graph
a.
b.
c.
*d.
e.
8. Identify the open intervals where the function
increasing or decreasing.
is
a. decreasing on
b. increasing on
c.
d. decreasing on
; increasing on
*e.
9. Identify the open intervals where the function
increasing or decreasing.
is
a. decreasing:
; increasing:
b. decreasing on
c. increasing:
; decreasing:
*d. increasing:
; decreasing:
e. increasing:
10. For the function
(a)
(b)
and
(c)
; decreasing:
:
Find the critical numbers of f (if any);
Find the open intervals where the function is increasing or decreasing;
Apply the First Derivative Test to identify all relative extrema.
Then use a graphing utility to confirm your results.
a. (a)
x = 0 ,
(b)
increasing:
(c)
relative max:
b. (a)
decreasing:
(c)
relative min:
increasing:
(c)
relative max:
; relative max:
; decreasing:
; no relative min.
x = 0 ,
(b)
increasing:
(c)
relative max:
e. (a)
; increasing:
x = 0 ,
(b)
*d. (a)
; relative min:
x = 0 ,
(b)
c. (a)
; decreasing:
; decreasing:
; relative min:
x = 0 ,
(b)
decreasing:
(c)
relative min:
; increasing:
; relative max:
11. The graph of f is shown in the figure. Sketch a graph of the derivative of
f.
a.
b.
c.
*d.
e. The derivative of f does not exist.
12. Determine the open intervals on which the graph of
concave downward or concave upward.
is
a. concave downward on
b. concave upward on
; concave downward on
*c. concave upward on
; concave downward on
d. concave downward on
e. concave downward on
; concave upward on
; concave upward on
13. Determine the open intervals on which the graph of
concave downward or concave upward.
is
a. concave downward on
b. concave downward on
; concave upward on
; concave upward on
c. concave upward on
; concave downward
on
d. concave downward on
; concave upward
on
*e. concave upward on
; concave downward
on
14. Find the points of inflection and discuss the concavity of the function.
a. inflection point at
; concave upward on
;
concave downward on
b. inflection point at
; concare downward on
;
concave upward on
c. inflection point at
; concave downward on
;
concave upward on
d. inflection point at
; concave downward on
;
concave upward on
*e. inflection point at
; concave upward on
downward on
15. Find the point of inflection of the graph of the function
on the interval
.
; concave
a.
b.
c.
*d.
e.
16. Find all relative extrema of the function
Second Derivative Test where applicable.
a. relative max:
. Use the
; no relative min
b. no relative max; no relative min
c. relative min:
*d. relative min:
e. relative min:
17. Match the function
; relative max:
; no relative max
; relative max:
with one of the following graphs.
a.
b.
*c.
d.
e.
18. Find the limit.
a.
b. 3
c.
d. –3
*e. 5
19. Find the limit.
a.
b. 1
*c. 0
d.
e.
20. Find the limit.
*a.
b.
c. 1
d. 7
e.
21. Sketch the graph of the function
intercepts, symmetry, and asymptotes.
using any extrema,
a.
b.
*c.
d.
e.
22. A model for the average typing speeds S (words per minute) of a typing
student after t weeks of lessons is given by
a. 96 words per minute
b. 12 words per minute
c. 56 words per minute
d. 162 words per minute
*e. 81 words per minute
Find
.
23. Analyze and sketch a graph of the function
a.
b.
*c.
.
d.
e.
24. Determine the slant asymptote of the graph of
a.
.
b.
c.
*d.
e. no slant asymptotes
25. Analyze and sketch a graph of the function
.
*a.
b.
c.
d.
e.
26. Use the following graph of
to sketch a graph of f.
a.
b.
c.
*d.
e.
27. The graph of f is shown below. For which value of x is
a.
b.
*c.
d.
e.
zero?
28. The graph of f is shown below. For which value of x is
minimum?
a.
b.
c.
*d.
e.
29. Find two positive numbers such that the sum of the first and twice the
second is 56 and whose product is a maximum.
a.
*b. 28 and 14
c.
d.
e.
30. Find the point on the graph of the function
that is
closest to the point
answer to four decimal places.
. Round all numerical values in your
*a.
b.
c.
d.
e.
31. Determine the dimensions of a rectangular solid (with a square base) with
maximum volume if its surface area is 529 square meters.
a. Dimensions:
b. Dimensions:
c. Dimensions:
*d. Dimensions:
e. Dimensions: